A fully data-driven method for estimating the shape of a point cloud

TL;DR

This paper introduces a data-driven method to estimate the shape of a point cloud's probability support, using an $r$-convex set estimator and a stochastic algorithm to select the shape parameter, applicable under flexible smoothness conditions.

Contribution

It proposes a novel, fully data-driven approach for estimating the support shape of a distribution using $r$-convex sets and an algorithm to select the shape parameter from data.

Findings

Achieves convergence rates comparable to convex hull estimators for convex sets.

Works under flexible smoothness conditions beyond convexity.

Demonstrates effectiveness through real data and simulations.

Abstract

Given a random sample of points from some unknown distribution, we propose a new data-driven method for estimating its probability support $S$. Under the mild assumption that $S$ is $r$-convex, the smallest $r$-convex set which contains the sample points is the natural estimator. The main problem for using this estimator in practice is that $r$ is an unknown geometric characteristic of the set $S$. A stochastic algorithm is proposed for selecting it from the data under the hypothesis that the sample is uniformly generated. The new data-driven reconstruction of $S$ is able to achieve the same convergence rates as the convex hull for estimating convex sets, but under a much more flexible smoothness shape condition. The practical performance of the estimator is illustrated through a real data example and a simulation study.